Numerical solutions of the Benjamin-Bona-Mahony-Burgers equation in one space dimension are considered using Crank-Nicolson-type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L ∞ -norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin-Bona-Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.

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In this article, a fully discrete Galerkin scheme based on a nonlinear Crank-Nicolson method to approximate the solution of the DGRLW equation is constructed. Some a priori bounds are proved as well as error estimates. Then, a linearized modification scheme by an extrapolation method is discussed. The two schemes are time second order convergence. The last part is devoted to some numerical results.